55 research outputs found
Diffusion systems and heat equations on networks
A theoretical framework for sesquilinear forms defined on the direct sum of
Hilbert spaces is developed in the first part. Conditions for the boundedness,
ellipticity and coercivity of the sesquilinear form are proved. A criterion of
E.-M. Ouhabaz is used in order to prove qualitative properties of the abstract
Cauchy problem having as generator the operator associated with the
sesquilinear form. In the second part we analyze quantum graphs as a special
case of forms on subspaces of the direct sum of Hilbert spaces. First, we set
up a framework for handling quantum graphs in the case of infinite networks.
Then, the operator associated with such systems is identified and investigated.
Finally, we turn our attention to symmetry properties of the associated
parabolic problem and we investigate the connection with the physical concept
of a gauge symmetry.Comment: 120 pages, PhD Thesi
Emergent Properties of Interacting Populations of Spiking Neurons
Dynamic neuronal networks are a key paradigm of increasing importance in brain research, concerned with the functional analysis of biological neuronal networks and, at the same time, with the synthesis of artificial brain-like systems. In this context, neuronal network models serve as mathematical tools to understand the function of brains, but they might as well develop into future tools for enhancing certain functions of our nervous system. Here, we present and discuss our recent achievements in developing multiplicative point processes into a viable mathematical framework for spiking network modeling. The perspective is that the dynamic behavior of these neuronal networks is faithfully reflected by a set of non-linear rate equations, describing all interactions on the population level. These equations are similar in structure to Lotka-Volterra equations, well known by their use in modeling predator-prey relations in population biology, but abundant applications to economic theory have also been described. We present a number of biologically relevant examples for spiking network function, which can be studied with the help of the aforementioned correspondence between spike trains and specific systems of non-linear coupled ordinary differential equations. We claim that, enabled by the use of multiplicative point processes, we can make essential contributions to a more thorough understanding of the dynamical properties of interacting neuronal populations
Parabolic systems with coupled boundary conditions
We consider elliptic operators with operator-valued coefficients and discuss
the associated parabolic problems. The unknowns are functions with values in a
Hilbert space . The system is equipped with a general class of coupled
boundary conditions of the form and
, where is
a closed subspace of . We discuss well-posedness and
further qualitative properties, systematically reducing features of the
parabolic system to operator-theoretical properties of the orthogonal
projection onto
Heat equation with dynamical boundary conditions of reactive-diffusive type
This paper deals with the heat equation posed in a bounded regular domain
coupled with a dynamical boundary condition of reactive-diffusive type,
involving the Laplace-Beltrami operator. We prove well-posedness of the problem
together with regularity of the solutions.Comment: 18 page
The Role of Inhibition in Generating and Controlling Parkinson’s Disease Oscillations in the Basal Ganglia
Movement disorders in Parkinson’s disease (PD) are commonly associated with slow oscillations and increased synchrony of neuronal activity in the basal ganglia. The neural mechanisms underlying this dynamic network dysfunction, however, are only poorly understood. Here, we show that the strength of inhibitory inputs from striatum to globus pallidus external (GPe) is a key parameter controlling oscillations in the basal ganglia. Specifically, the increase in striatal activity observed in PD is sufficient to unleash the oscillations in the basal ganglia. This finding allows us to propose a unified explanation for different phenomena: absence of oscillation in the healthy state of the basal ganglia, oscillations in dopamine-depleted state and quenching of oscillations under deep-brain-stimulation (DBS). These novel insights help us to better understand and optimize the function of DBS protocols. Furthermore, studying the model behavior under transient increase of activity of the striatal neurons projecting to the indirect pathway, we are able to account for both motor impairment in PD patients and for reduced response inhibition in DBS implanted patients
Effect of network structure on spike train correlations in networks of integrate-and-fire neurons
Well-Posedness and Symmetries of Strongly Coupled Network Equations
We consider a diffusion process on the edges of a finite network and allow
for feedback effects between different, possibly non-adjacent edges. This
generalizes the setting that is common in the literature, where the only
considered interactions take place at the boundary, i. e., in the nodes of the
network. We discuss well-posedness of the associated initial value problem as
well as contractivity and positivity properties of its solutions. Finally, we
discuss qualitative properties that can be formulated in terms of invariance of
linear subspaces of the state space, i. e., of symmetries of the associated
physical system. Applications to a neurobiological model as well as to a system
of linear Schroedinger equations on a quantum graph are discussed.Comment: 25 pages. Corrected typos and minor change
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